skull conductivity Errors AND
the Inverse problem of electroencephalograpy

Stefan Finke¹, Ramesh M. Gulrajani¹, Jean Gotman^2
1Institut de génie biomédical, Université de Montréal,
C.P. 6128, succursale Centre-ville, Montréal, Québec, CANADA H3C 3J7
²Montreal Neurological Institute, McGill University
3801 University Street, Montreal, Quebec, CANADA, H3A 2B4

Abstract: Forward transfer matrices relating dipole source to surface potentials can be determined via conventional or reciprocal approaches. In numerical simulations with a triangulated boundary-element three-concentric-spheres head model, we compare two inverse EEG solutions, those obtained with conventional and those with reciprocal transfer matrices. Dipole localization errors are presented in both cases for varying dipole eccentricity and with incorrect values for skull conductivity. In the presence of noise, both approaches were found to be equally robust to skull conductivity errors.

INTRODUCTION

There has been much recent interest in a reciprocal approach to the inverse problem of electroencephalography (EEG), wherein the forward transfer matrix relating dipole source to scalp potentials is obtained via a reciprocal approach. This reciprocal approach first entails calculating the electric field that results at the dipole location within the brain, due to current injection and withdrawal at the electrode sites [1]. The forward transfer coefficients are then simply obtained from the scalar product of this electric field and the dipole moment. One advantage of a reciprocally-computed transfer matrix is that the volume conductor discretization can be refined at the known scalp electrode sites, presumably increasing the precision of the transfer coefficients and hence that of the computed inverse dipole. In the usual conventional approach, the transfer coefficient between source dipole and electrode site is obtained via an integral equation. Added precision is difficult since the region to be refined is near the dipole, whose position is unknown.

We describe herein inverse EEG solutions obtained using reciprocal transfer matrices employing a boundary-element volume conductor in which only interfaces between regions of assumed homogeneity and isotropy are discretized. Earlier, we found little difference in precision in reciprocally-computed transfer matrices over conventionally-computed ones [2]. However, the reciprocal transfer matrix was more robust to changes in skull conductivity. Here, we see if this robustness also translates to a greater invariance of inverse EEG solutions employing reciprocal transfer matrices, in the presence of errors in relative skull conductivity.

METHODS

The three-concentric-spheres head model was employed with relative conductivities for the scalp, skull, and brain of 1, 1/15, and 1, respectively [3]. Sphere radii were 10 cm, 9.2 cm, and 8.7 cm. The spheres were discretized into planar triangles employing regularly spaced lines of latitude and longitude, except around 42 electrode sites where a higher triangle density was used. With the conventional vertex (CV) approach, one of the vertices of the innermost triangle in each high density region was selected as the electrode site and a linear variation in potential was assumed on each triangle. A slightly different discretization was needed with the reciprocal vertex (RV) approach. Because of the difficulty of calculating the current distribution following injection at a vertex, a curvilinear quadrilateral innermost element was used, with the site of current injection at the center of the quadrilateral. There were 2228 triangles per sphere for the CV approach, and 2480 triangles plus 42 curvilinear quadrilaterals per sphere for the RV approach.

Analytic potentials, due to tangential and radial current dipoles at different eccentricities in the inner sphere, were calculated assuming no noise, and then perturbed by the addition of 10% or 20% noise at the 41 electrode sites to be used for inverse computations. The 42nd electrode at the summit of the head model was used as a reference. These noise levels are representative of the noise to be expected during EEG measurement (for EEG spikes, the noise level typically equals 20% but for averaged spikes, it typically equals 10%). The noisy analytic potentials served as inputs for the inverse dipole solution.

 The simplex algorithm [4] was used to select the best position for the inverse dipole by minimizing the relative difference error measure (RDM) defined as

where  and  denote the analytic and inverse-dipole potentials, respectively. Furthermore, a large value was added to the RDM for dipole positions outside the brain, thereby restricting the inverse solution to within the brain. Note that the simplex algorithm only searches for the three position coordinates of the dipole, since the dipole moment at each position is obtained from the “normal equations” resulting from a linear least-squares minimization of the RDM.

For each inverse solution, 10 simplex minimizations with different randomly chosen starting points were run, with stopping points either when the difference in RDM between successive simplex iterations dropped below 0.0001 or following a maximum of 1000 iterations. The computed dipole  position  was  obtained   from  the  simplex   with   the lowest RDM among the set of converging simplexes. The distance between the dipole computed by this simplex and the true dipole was deemed the position error.

RESULTS

The position errors of inversely-computed dipoles from potentials due to tangential source dipoles are shown in Fig. 1 for different source dipole eccentricities. An error of  in relative skull conductivity was introduced when estimating the inverse dipole. Here, the RV approach appears somewhat more invariant to errors in skull conductivity than the CV approach. In Fig. 2, the simulations were identical but for the presence of 10% noise in the input potentials. At this noise level, conventional and reciprocal approaches both worked equally well for tangential dipoles. The effect of skull conductivity errors was similar for a 20% noise level.

Figure 1.  Dipole position error plotted against dipole eccentricity for tangential dipoles in the absence of noise and with 0% and  error in relative skull conductivity. The CV approach is shown in (a) and the RV approach in (b).

Figure 2.  Dipole position error plotted against dipole eccentricity for tangential dipoles with 10% noise and with 0% and  error in relative skull conductivity. The CV approach is shown in (a) and the RV approach in (b).

DISCUSSION

In the absence of noise, reciprocal transfer coefficients result in inverse solutions that are more resistant to incorrect estimates of skull conductivity than those computed with conventional transfer coefficients, as suggested by the relative invariance of reciprocal transfer coefficient matrices to changes in skull conductivity [2]. However, in the presence of Gaussian noise in the surface potentials,  both conventional and reciprocal transfer coefficients yielded inverse solutions of equal robustness to skull conductivity errors.

Acknowledgments:  Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Canadian Institutes of Health Research. S. Finke is the recipient of an MD/PhD Scholarship from the Canadian Institutes of Health Research and is also supported in part by le Fonds de la recherche en santé du Québec.

REFERENCES

[1] D.J. Fletcher, A. Amir, D.L. Jewett, and G. Fein, “Improved method for computation of potentials in a realistic head shape model,” IEEE Trans. Biomed. Eng., vol. 42, pp. 1094-1104, 1995.

[2] S. Finke   and   R.M. Gulrajani,   “Conventional   and   reciprocal  approaches to the forward problem of electroencephalography,” Electromagnetics [Special issue on Bioelectromagnetic Forward and Inverse Problems], vol. 21, pp. 513-530, 2001.

[3] T.F. Oostendorp, J. Delbeke, and D.F. Stegeman, “The conductivity of the human skull: Results of in vivo and in vitro measurements,” IEEE Trans. Biomed. Eng., vol. 47, pp. 1487-1492, 2000.

[4] B. He, T. Musha, Y. Okamoto, S. Homma, Y. Nakajima, and T. Sato, “Electric dipole tracing in the brain by means of the boundary element method and its accuracy,” IEEE Trans. Biomed. Eng., vol. 34, pp. 406-414, 1987.